Containment and Evasion in Stochastic Point Data

  • Nirman KumarEmail author
  • Subhash Suri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


Given two disjoint and finite point sets \(A\) and \(B\) in \(\mathrm{I\! R}^d\), we say that \(B\) is contained in \(A\) if all the points of \(B\) lie within the convex hull of \(A\), and that \(B\) evades \(A\) if no point of \(B\) lies inside the convex hull of \(A\). We investigate the containment and evasion problems of this type when the set \(A\) is stochastic, meaning each of its points \(a_i\) is present with an independent probability \(\pi (a_i)\). Our model is motivated by situations in which there is uncertainty about the set \(A\), for instance, due to randomized strategy of an adversarial agent or scheduling of monitoring sensors. Our main results include the following: (1) we can compute the exact probability of containment or evasion in two dimensions in worst-case \(O(n^4 + m^2)\) time and \(O(n^2 + m^2)\) space, where \(n = | A | \) and \(m = | B | \), and (2) we prove that these problems are #P-hard in 3 or higher dimensions.


Convex Hull Dynamic Programming Algorithm Membership Query Triangle Emptiness Evasion Problem 
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Supplementary material


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta-BarbaraUSA

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